We study the semigroup $\textbf{ { ID}}_{ \infty}$ of all partial isometries of the set of integers $\mathbb{ Z}$. It is proved that the quotient semigroup $\textbf{ { ID}}_{ \infty}/\mathfrak{ C}_{ \textsf{ mg}}$, where $\mathfrak{ C}_{ \textsf{ mg}}$ is the minimum group congruence, is isomorphic to the group ${ \textsf{ Iso}}(\mathbb{ Z})$ of all isometries of $\mathbb{ Z}$, $\textbf{ { ID}}_{ \infty}$ is an $F$-inverse semigroup, and $\textbf{ { ID}}_{ \infty}$ is isomorphic to the semidirect product ${ \textsf{ Iso}}(\mathbb{ Z})\ltimes_\mathfrak{ h}\mathscr{ P}_{ \!\infty}(\mathbb{ Z})$ of the free semilattice with unit $(\mathscr{ P}_{ \!\infty}(\mathbb{ Z}),\cup)$ by the group ${ \textsf{ Iso}}(\mathbb{ Z})$. We give the sufficient conditions on a shift-continuous topology $\tau$ on $\textbf{ { ID}}_{ \infty}$ when $\tau$ is discrete. A non-discrete Hausdorff semigroup topology on $\textbf{ { ID}}_{ \infty}$ is constructed. Also, the problem of an embedding of the discrete semigroup $\textbf{ { ID}}_{ \infty}$ into Hausdorff compact-like topological semigroups is studied.
Categories: math.GR